Mohr’s circle for combined stress:\nFor a body under normal stresses σx and σy (mutually perpendicular) and shear stress τxy, what is the radius of Mohr’s circle?

Introduction / Context:
Mohr’s circle is a graphical representation of plane stress transformation. Its center and radius encode the entire 2D stress state, allowing quick determination of principal stresses, maximum shear stress, and stresses on any inclined plane.

Given Data / Assumptions:

• Plane stress with components σx, σy, and τxy on orthogonal faces.
• Linear elasticity; sign conventions per standard strength of materials.
• Seeking the radius (R) of the Mohr’s circle.

Concept / Approach:
The coordinates of two diametrically opposite points on Mohr’s circle correspond to the stress states on perpendicular planes. The circle’s center C lies at ( (σx + σy)/2 , 0 ), and the distance from C to either point equals the radius R. Using the distance formula yields the standard expression for R.

Step-by-Step Solution:

Verification / Alternative check:
Principal stresses are σ1,2 = (σx + σy)/2 ± R. Substituting R from the derived formula reproduces the analytical transformation equations for principal values and maximum shear = R.

Why Other Options Are Wrong:

• (σx + σy)/2 is the center, not the radius.
• √(σx^2 + σy^2 + τxy^2) does not reflect geometric derivation and is dimensionally inconsistent for transformation.
• (σx − σy)/τxy and τxy/2 do not have the correct form or units for a radius.

Common Pitfalls:
Mixing up center and radius; incorrect sign for τxy; forgetting that Mohr’s rotation uses 2θ.

Final Answer:
R = √[ ((σx − σy) / 2)^2 + (τxy)^2 ]